PHYSICS

I. I. MOISEEV-OLKHOVSKII

ON A PLANE LINEAR PROBLEM OF GENERALIZED HYDRODYNAMICS*

(Presented by Academician N. N. Bogolyubov on 19 VI 1957)

1. In the paper ((^{1})) it was shown that the Boltzmann kinetic equation is the first approximation in the solution of the “Bogolyubov chain” with respect to the quantity (n/n_0) ((n) is the number density of particles, (n_0 = 1/r_0^3), where (r_0) is the constant of short-range action), while no restrictions are imposed on the ratio of the relaxation time (\Delta t_p) to the time interval (\Delta t) characteristic for the process under consideration. Consequently, it is legitimate to pose the question of such a method for solving the Boltzmann equation which would not impose restrictions on (\Delta t_p/\Delta t). From this point of view, let us consider the method of “moments.” To this end we shall obtain equations for the moments of the single-particle distribution function, without using Maxwell’s transport equations or the conservation equations of a continuous medium, as was done in ((^{2})), but proceeding only from the Boltzmann equation and the known normalization conditions. We write the Boltzmann equation and the normalization conditions in terms of the dimensionless distribution function (g) and the dimensionless relative velocity of molecules (\vec{\xi}):

[
\frac{dg}{dt}+c_e\xi_i\frac{\partial g}{\partial x_i}
+\frac{1}{c_e}\frac{\partial g}{\partial \xi_i}\left(X_i-\frac{du_i}{dt}\right)
-\xi_j\frac{\partial g}{\partial \xi_j}\left(\frac{d\ln c_e}{dt}
+c_e\xi_i\frac{\partial\ln c_e}{\partial x_i}\right)
-\xi_i\frac{\partial g}{\partial \xi_j}\frac{\partial u_j}{\partial x_i}
+g\left[\frac{d\ln(n/c_e^3)}{dt}
+c_e\xi_i\frac{\partial\ln(n/c_e^3)}{\partial x_i}\right]
=nJ(g,g_1);
\tag{1}
]

[
\int g\,d\vec{\xi}=1;\qquad
\int \xi_i g\,d\vec{\xi}=0;\qquad
\int \xi^2 g\,d\vec{\xi}=3;\qquad
P_{ij}=\rho c_e^2\int \xi_i\xi_j g\,d\vec{\xi};
]

[
q_i=\frac{\rho c_e^3}{2}\int \xi_i\xi^2 g\,d\vec{\xi};\qquad
S_{ijk}=\rho c_e^3\int \xi_i\xi_j\xi_k g\,d\vec{\xi}
\quad \text{etc.,}
]

where

[
g(t,\mathbf r,\vec{\xi})=f\,\frac{c_e^3(t,\mathbf r)}{n(t,\mathbf r)};
\qquad
\vec{\xi}=\frac{\mathbf c}{c_e};
\qquad
\xi^2=\xi_1^2+\xi_2^2+\xi_3^2;
]

[
c_e=\sqrt{\frac{2}{3}\,e(t,\mathbf r)};
\qquad
e(t,\mathbf r)=\frac{1}{\rho}\int \frac{mc^2}{2}f\,d\mathbf c;
\qquad
\frac{d}{dt}=\frac{\partial}{\partial t}+u_i\frac{\partial}{\partial x_i};
]

[
J(gg_1)=\int c_e|\vec{\xi}_1-\vec{\xi}|{g'g_1'-gg_1}\,b\,db\,d\varphi\,d\vec{\xi}_1.
]

Substitute in (1) the expansion of (g) in generalized Hermite polynomials (H^{(r)}(\vec{\xi})) ((^{3}))

[
g=g_0\sum_{r=0}^{\infty}\ \sum_{i_1\ldots i_r}\frac{1}{r!}\alpha^{(r)}{i_1\ldots i_r}H^{(r)},
\tag{2}
]

* Reported at the All-Union Acoustics Conference on 25 VI 1957.

where (g_0=(2\pi)^{-3/2}\exp(-\xi^2/2)); (\alpha^{(r)}_{i_1\ldots i_r}) are unknown functions of (t,\mathbf r). (By virtue of the orthonormality of the functions (H^{(r)}), the (\alpha^{(r)}) are related to the moments of the function (g).) We obtain*:

[
\begin{aligned}
&\frac{d\alpha^{(s)}{j_1\ldots j_s}}{dt}
+c_e\frac{\partial}{\partial x\gamma}\left{
\alpha^{(s+1)}{\gamma j_1\ldots j_s}
+\sum}}\delta_{\gamma j_1}\alpha^{(s-1){j_2\ldots j_s}
\right}
\
&\quad
+\frac{1}{c_e}\left(\frac{du\gamma}{dt}-X_\gamma\right)
\sum_{\hat P_{j_1\ldots j_s}}\delta_{\gamma j_1}\alpha^{(s-1)}{j_2\ldots j_s}
\
&\quad
+\frac{d\ln(nc_e^s)}{dt}\,\alpha^{(s)}
+2\frac{d\ln c_e}{dt}
\sum_{\hat P_{j_1\ldots j_s}}\delta_{j_1j_2}\alpha^{(s-2)}{j_3\ldots j_s}
\
&\quad
+c_e\frac{\partial\ln(nc_e^{s+1})}{\partial x\gamma}
\left{
\alpha^{(s+1)}{\gamma j_1\ldots j_s}
+\sum}}\delta_{\gamma j_1}\alpha^{(s-1){j_2\ldots j_s}
\right}
\
&\quad
+2\frac{\partial c_e}{\partial x\gamma}
\left{
\sum_{\hat P_{j_1\ldots j_s}}\delta_{j_1j_2}\alpha^{(s-1)}{\gamma j_3\ldots j_s}
+
\sum}}\delta_{j_1j_2
\sum_{\hat P_{j_3\ldots j_s}}\delta_{\gamma j_3}\alpha^{(s-3)}{j_4\ldots j_s}
\right}
\
&\quad
+\frac{\partial u\gamma}{\partial x_\gamma}\,\alpha^{(s)}{j_1\ldots j_s}
+\frac{\partial u\gamma}{\partial x_\nu}
\left{
\sum_{\hat P_{j_1\ldots j_s}}\delta_{\gamma j_1}\alpha^{(s)}{\nu j_2\ldots j_s}
+
\sum}}\delta_{\gamma j_1
\sum_{\hat P_{j_2\ldots j_s}}\delta_{\gamma j_2}\alpha^{(s-2)}{j_3\ldots j_s}
\right}
=J^{(s)},
\end{aligned}
\tag{3}
]

where

[
J^{(s)}{j_1\ldots j_s}=n\int H^{(s)}.}J(gg_1)\,d\vec{\xi
]

Putting (s=0,1,2,\ldots) and taking into account the relation of (\alpha^{(s)}) to the moments, we obtain equations for the moments:

[
\frac{dn}{dt}+\frac{\partial nu_\gamma}{\partial x_\gamma}=0,
\qquad
\frac{\partial u_{j_1}}{\partial t}+u_\gamma\frac{\partial u_{j_1}}{\partial x_\gamma}
=X_{j_1}-\frac{1}{\rho}\frac{\partial P_{\gamma j_1}}{\partial x_\gamma};
]

[
\frac{\partial P_{j_1j_2}}{\partial t}
+u_\gamma\frac{\partial P_{j_1j_2}}{\partial x_\gamma}
+\frac{\partial S_{\gamma j_1j_2}}{\partial x_\gamma}
+\frac{\partial u_\gamma}{\partial x_\gamma}P_{j_1j_2}
+\frac{\partial u_{j_1}}{\partial x_\gamma}P_{\gamma j_2}
+\frac{\partial u_{j_2}}{\partial x_\gamma}P_{\gamma j_1}
=\rho c_e^2J^{(2)}_{j_1j_2},
\tag{4}
]

[
\begin{aligned}
&\frac{\partial S_{j_1j_2j_3}}{\partial t}
+u_\gamma\frac{\partial S_{j_1j_2j_3}}{\partial x_\gamma}
+\frac{\partial u_\gamma}{\partial x_\gamma}S_{j_1j_2j_3}
+\sum_{\hat P_{j_1j_2j_3}}\frac{\partial u_{j_1}}{\partial x_\gamma}S_{\gamma j_2j_3}
-\frac{1}{\rho}\sum_{\hat P_{j_1j_2j_3}}P_{j_1j_2}\frac{\partial P_{\gamma j_3}}{\partial x_\gamma}
\
&\quad
+\frac{\partial}{\partial x_\gamma}
\sum_{\hat P_{j_1j_2j_3}}
\left{c_e^2P_{\gamma j_1}\delta_{j_2j_3}
+c_e^2P_{j_2j_3}\delta_{\gamma j_1}
-\rho c_e^4\delta_{\gamma j_1}\delta_{j_2j_3}\right}
+\frac{\partial}{\partial x_\gamma}{\rho c_e^4\alpha^{(4)}{\gamma j_1j_2j_3}}
=\rho c_e^3J^{(3)}.
\end{aligned}
]

Restriction to a finite number of moments imposes only the requirement
(\alpha^{(r)}{j_1\ldots j_r}\big|\ll1). It follows from the derivation that the method of moments does not require any restriction on the magnitude of (\Delta t_p/\Delta t). In this connection we shall call the system (4) the equations of generalized hydrodynamics, valid for describing rapid processes.

* Here and below, (\hat P_{j_1\ldots j_s}) denotes summation over the terms, distinct from one another, obtained as a result of permutation of the indices.

** The calculation of (J^{(2)}{j_1j_2}) and (J^{(3)}}) from (4) was carried out in (2). In (5) we have restricted ourselves to terms linear in (J^{(2){j_1j_2}) and (J^{(3)}_1) depends on (T), (m), and the law of interaction between the particles.}) with respect to (\alpha^{(s)}). (B^{(2)

1. To solve the plane linear problem of the propagation of small perturbations with allowance for the processes of transfer of momentum and energy, six quantities are necessary: (n,\ u_1,\ p_{11},\ p_{22}+p_{33},\ S_{111},\ S_{122}+S_{123}), or (n,\ u_1,\ p_{11},\ p=\frac13(p_{11}+p_{22}+p_{33}),\ S_{111},\ S_1=S_{111}+S_{122}+S_{133}). For small deviations from the equilibrium stationary state, in the case considered, from (4) we obtain

[
\frac{\partial \beta^{(0)}}{\partial t}+(c_\varepsilon)0\frac{\partial \beta^{(1)}_1}{\partial x_1}=0;\qquad
\frac{\partial \beta^{(1)}_1}{\partial t}+(c\varepsilon)0\frac{\partial \beta^{(2)}=0;}}{\partial x_1
]

[
\frac{\partial \beta^{(2)}{11}}{\partial t}
+3(c\varepsilon)0\frac{\partial \beta^{(1)}_1}{\partial x_1}
+(c\varepsilon)0\frac{\partial \beta^{(3)}}}{\partial x_1
=6(\eta)0B^{(2)}_1\left(\beta^{(2)}-\beta^{(2)}\right);
]

[
\frac{\partial \beta^{(2)}}{\partial t}
+\frac53(c_\varepsilon)0\frac{\partial \beta^{(1)}_1}{\partial x_1}
+\frac13(c\varepsilon)_0\frac{\partial \beta^{(3)}_1}{\partial x_1}=0;
\tag{5}
]

[
\frac{\partial \beta^{(3)}{111}}{\partial t}
-3(c\varepsilon)0\frac{\partial \beta^{(0)}}{\partial x_1}
+3(c\varepsilon)0\frac{\partial \beta^{(2)}}}{\partial x_1
=3(\eta)0B^{(2)}_1\left(\beta^{(3)}_1-3\beta^{(3)}\right);
]

[
\frac{\partial \beta^{(3)}1}{\partial t}
-5(c\varepsilon)0\frac{\partial \beta^{(0)}}{\partial x_1}
+2(c\varepsilon)0\frac{\partial \beta^{(2)}}}{\partial x_1
+3(c_\varepsilon)_0\frac{\partial \beta^{(2)}}{\partial x_1}
=-4(\eta)_0B^{(2)}_1\beta^{(3)}_1,
]

Table 1

Dimensionless velocities (V/V_0) and attenuations per mean free path (\varkappa l) as functions of (\varepsilon=\nu\mu/p) (\left( V_0=V_1\big|{\varepsilon\to0};\ l=\dfrac{8}{15\sqrt{2\pi}}\dfrac{c\varepsilon}{nB^{(2)}_1}\right))

where
[
\beta^{(0)}=\Delta n/(n)0,\qquad
\beta^{(1)}_1=\Delta u_1/(c\varepsilon)0;
]
[
\beta^{(2)}/(\rho c_\varepsilon^2)}=\Delta p_{110,\qquad
\beta^{(2)}=\Delta p/(\rho c\varepsilon^2)0;
]
[
\beta^{(3)}/(\rho c_\varepsilon^3)}=\Delta S_{1110,\qquad
\beta^{(3)}_1=\Delta S_1(\rho c\varepsilon^3)_0.
]

To equation (5), in the case of a prescribed frequency, there corresponds the dispersion equation

[
{[1-\tfrac23\cdot 19(\pi\varepsilon)^2]+i[\tfrac{57}{9}(\pi\varepsilon)-8(\pi\varepsilon)^3]}u^6
+{[-\tfrac53+\tfrac{172}{3}(\pi\varepsilon)^2]+i[-\tfrac{164}{9}(\pi\varepsilon)+56(\pi\varepsilon)^3]}u^4
]
[
+{[-\tfrac{98}{3}(\pi\varepsilon)^2]+i[5(\pi\varepsilon)-72(\pi\varepsilon)^3]}u^2
+i[24(\pi\varepsilon)^3]=0,
\tag{6}
]

where
[
u=\omega/k(c_\varepsilon)0;\qquad
(c\varepsilon)_0=\sqrt{kT/m};
]
[
\varepsilon=\nu/6nB^{(2)}_1=\nu\mu/p;\quad
\nu\text{ is the wave frequency;}\quad
p\text{ is the gas pressure;}
]
[
\mu\text{ is the first approximation of the Chapman viscosity coefficient.}
]
The results of the numerical solution of (6) are given in Table 1.

If (\varepsilon=\nu\,\dfrac{\mu}{p}\ll1), then (6), with accuracy up to (\varepsilon^2), gives

[
V_1=V_0\left[1+4.3\pi^2\left(\frac{\nu\mu}{p}\right)^2\right];\qquad
\varkappa_1=\frac{42\pi^2}{5\sqrt{15}}\frac{\nu^2\mu}{p}\sqrt{\frac{m}{kT}}
=\frac76\,\frac{\omega^2\mu}{\rho V_0^3};
\tag{7}
]

[
V_2=\sqrt{\frac{6\pi\nu\mu kT}{pm}}
\left[1-1.3\pi\left(\frac{\nu\mu}{p}\right)\right];\qquad
\varkappa_2=\sqrt{\frac{2\pi\nu pm}{3\mu kT}}
\left[1-1.3\pi\left(\frac{\nu\mu}{p}\right)\right].
\tag{8}
]

From (7) and (8), neglecting the second terms in the brackets, we obtain the well-known expressions for the adiabatic speed of sound and for the velocity and attenuation of “thermal waves” ((^4)):

[
V_0=\sqrt{\frac53\,\frac{kT}{m}};\qquad
(V_2)_0=\sqrt{2\chi\omega};\qquad
(\varkappa_2)_0=\sqrt{\frac{\omega}{2\chi}};\qquad
\chi=\frac32\,\frac{\mu}{\rho}=\frac{\lambda}{\rho c_p},
]

where (\chi) is the thermal diffusivity. The attenuation of sound (x_1) coincides with the known expression ({}^{(4)}) when the first viscosity and thermal conductivity are taken into account. From consideration of the numerical solution of (6) and (7), (8), it follows: the first solution—the “acoustic branch”—gives the translational dispersion of sound ({}^{(5-8)}) and describes the boundary of propagation of ultrasound in monatomic gases as a function of various parameters ({}^{(9)}); the second solution gives the same complete description of “thermal waves”; the third solution, apparently, is practically not realized because of the large values of (x l_3). A comparison of the first solution with experiment is given in Fig. 1 in the notation of ({}^{(7)}). For brevity of presentation, the graph is given only for the dimensionless velocity. In ({}^{(7)}) there is a quantitative discrepancy between experiment and the theories of Navier—Stokes, Burnett, and the “super”-Burnett approximation in the region (r \lesssim 1) or (\varepsilon \gtrsim 0.16) ((r = 1/2\pi\varepsilon)). In fact, the discrepancy is qualitative, since the author uses dispersion equations containing terms of higher order in (1/r) (or in (\varepsilon)) in comparison with the original hydrodynamic equations. In the theory being developed, this incorrectness is absent, and for the speed of sound the theory agrees with experiment over the entire interval of values of (\varepsilon) (or (r)). The conclusions of the present work are valid for monatomic gases in the Boltzmann approximation.

Fig. 1

In conclusion I express my gratitude to Acad. N. N. Bogolyubov for discussion of the results, and also to T. M. Cherkasova for assistance in the numerical calculations.

Moscow State University
named after M. V. Lomonosov

Received
2 VII 1957

CITED LITERATURE

({}^{1}) N. N. Bogolyubov, Problems of Dynamical Theory in Statistical Physics, Moscow, 1946.
({}^{2}) H. Grad, Comm. on Pure and Appl. Math., 2, No. 4, 331 (1949).
({}^{3}) H. Grad, Comm. on Pure and Appl. Math., 2, No. 4, 325 (1949).
({}^{4}) L. D. Landau, E. M. Lifshitz, Mechanics of Continuous Media, 1954.
({}^{5}) H. Primakoff, J. Acoust. Soc. Am., 13, 15 (1942).
({}^{6}) H. S. Tsien, R. Schamberg, J. Acoust. Soc. Am., 18, 334 (1946).
({}^{7}) M. Greenspan, J. Acoust. Soc. Am., 22, 568 (1950); 28, 644 (1956).
({}^{8}) R. Boyer, J. Acoust. Soc. Am., 23, 176 (1951).
({}^{9}) I. I. Monseev-Olkhovskii, ZhETF, 31, 238 (1956).